I want to prove that there exist projective covers in the category of graded modules over an algebra.
I am fairly new to "this" kind of mathematic and don't really know where to start: I found the answer to this question "graded modules have enough projectives" and thought this would be a first step, since it provides the existence of a projective graded module and a surjective map for every graded module. Now I have to get to the superfluousness of the kernel somehow – the constructed graded-free module $\oplus_i R[-i]^{\oplus B_{-i}}$ might be too large; and even if not, I have no idea how to relate the graded structure to the superfluos kernel.
And since there is no question mark in my question so far: Can anyone give me any hints on how to prove this or how to continue my "thoughts"?